Recent deep Hubble Space Telescope WFC3 imaging suggests that a majority of compact quiescent massive galaxies at z ~ 2 may contain disks. To investigate this claim, we have compared the ellipticity distribution of 31 carefully selected high-redshift massive quiescent compact galaxies to a set of mass-selected ellipticity and Sérsic index distributions obtained from two-dimensional structural fits to ~40, 000 nearby galaxies from the Sloan Digital Sky Survey. A Kolmogorov-Smirnov test shows that the distribution of ellipticities for the high-redshift galaxies is consistent with the ellipticity distribution of a similarly chosen sample of massive early-type galaxies. However, the distribution of Sérsic indices for the high-redshift sample is inconsistent with that of local early-type galaxies, and instead resembles that of local disk-dominated populations. The mismatch between the properties of high-redshift compact galaxies and those of both local early-type and disk-dominated systems leads us to conclude that the basic structures of high-redshift compact galaxies probably do not closely resemble those of any single local galaxy population. Any galaxy population analog to the high-redshift compact galaxies that exists at the current epoch is either a mix of different types of galaxies, or possibly a unique class of objects on their own.